A383243 Primes of the form p(k)*p(k+1)*(p(k+1) - p(k)) - 1 sorted by increasing k.

5, 29, 307, 883, 1747, 4001, 6067, 26227, 108883, 152083, 424481, 311347, 396883, 848201, 580627, 1713709, 1814509, 864883, 5092973, 3046789, 3386989, 1664083, 2581961, 2196307, 2304307, 2377747, 6955309, 3526867, 4088467, 20916053, 4796083, 7339361

Offset: 1

Author

Clark Kimberling, May 07 2025

Keywords

• nonn

Comments

Conjecture: there are infinitely many such primes.

Crossrefs

Cf. A000042, A383242, A383244.

Primes in A383241.

Programs

Maple

q:= 2; R:= NULL: count:= 0:
while count < 100 do
  p:= q;
  q:= nextprime(q);
  v:= p*q*(q-p)-1;
  if isprime(v) then R:= R,v; count:= count+1 fi;
od:
R; # Robert Israel, May 11 2025

Mathematica

z = 200; p[n_] := Prime[n];
f[n_] := p[n]*p[n + 1]*(p[n + 1] - p[n])
t1 = Table[f[n] - 1, {n, 1, z}];    (* A383241 *)
t2 = Table[f[n] + 1, {n, 1, z}];    (* A383242 *)
Select[t1, PrimeQ[#] &]  (* A383243 *)
Select[t2, PrimeQ[#] &]  (* A383244 *)